The measure of drought duration strongly depends on the definition considered. In meteorology, dryness is habitually measured by means of fixed thresholds (e.g. 0.1 or 1 mm usually define dry spells) or climatic mean values (as is the case of the standardised precipitation index), but this also depends on the aggregation time interval considered. However, robust measurements of drought duration are required for analysing the statistical significance of possible changes. Herein we climatically classified the drought duration around the world according to its similarity to the voids of the Cantor set. Dryness time structure can be concisely measured by the

Drought depends mainly on the sector affected and the timescale considered

A commonly used method for measuring the dimension of fractal objects involves

Another related measure involves the Lyapunov exponent, which indicates the rate of separation of infinitesimally close trajectories, or involves the inverse, sometimes referred to as Lyapunov time, since it indicates the time expected to become a chaotic trajectory

The fractal behaviour of dry spells can be observed in Richardson's log–log plot of cumulative dry durations with regard to different unit durations

According to a multifractal analysis of the standardised precipitation, power-law decay distribution describes well the probability density function of return intervals of drought events

In addition, the fractal density of wet (or dry) spells can be estimated according to the

The main fractal measure was estimated for dry-spell density by means of the

In a similar way to precipitation, the maximum accumulated dry-spell duration (

Due to the low probability of the longest spells, a high maximum duration

The Akaike information criterion (AIC) was applied to each fitted model using the log-likelihood function according to the equation

Finally, the lacunarity of the Cantor set was compared with the frequency distribution of dry-spell durations for a given time series of

The value of the iteration

The results of the dry-spell

The data used in this study were obtained from the full global gridded daily Multi-Source Weighted-Ensemble Precipitation (MSWEP) dataset

By analysing dry-spell spells (DSSs), the first overview of the spatial distribution of the DSS

Spatial distribution of regularity of the lacunarity, averaging for all DSS events:

More specifically, rainforests like those in the Amazon, the Congo, or southeastern Asia present values of

High values of the index (

Spatial distribution of other fractal measures applied to dry spells:

Three main sets can be identified according to the predominance of low (L), medium (M), or high (H) values of the DSS

Climatic classification of meteorological droughts around the world: regions with low (L), medium (M), or high (H) values of the DSS

Since the total (1-D) length of the Cantor set is zero, the total length of the complementary gaps is equal to 1. That is, following the analogy between the drought duration and the Cantor set lacunarity, the total duration of a dry-spell series approaches 1 when the size of the measurement box is accurate. For instance, one can find dry days in a wet month, and on rainy days, there can be several hours with no rain. If a ground point is used for measurement, the duration of a raindrop hitting the ground (from leading surface to trailing surface) tends towards zero, and thus the dry pauses are distributed paradoxically throughout an entire rainy day.

According to this idea, the dimension of the drought duration is practically 1 (the length of the time series), and the box-counting dimension therefore makes more sense for measuring rainfall duration than for estimating drought duration. However, the lacunarity of the drought can be analysed by means of other measures, such as the Gini index (

The DSS

In short, the time patterns of the dryness of a climate can be characterised in a simple manner by means of the

As in (fractal) wet spells, the behaviour of dryness is self-similar on all timescales – that is to say, dry spells can be used at the daily, monthly, yearly resolution, etc., considering specific dryness thresholds. This is guaranteed by the goodness of fit of the

Finally, the main limitation in this study is the possible errors derived from the used precipitation dataset. However, the source errors are generally smaller for the breaks between dry and wet spells because they are not influenced by the absolute precipitation amount.

The datasets from Figs. 1–3 and Fig. A1 can be accessed through

As a principal conclusion, the study demonstrates that drought lacunarity can be analysed with the use of self-similarity features obtained from the DSS

Consequently, we arrived at a second conclusion: the methodology developed can be used to classify typologies of meteorological drought. Indeed, six climate types have been proposed: these result from the combination of three classes of DSS

The Cantor-based exponent and the

The authors declare that they have no conflict of interest.

We are grateful for the support provided by the RESCCUE project, which received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 700174). We also wish to acknowledge the support received from the Spanish projects CGL2017-83866-C3-2-R and Climatology Group 2017 SGR 1362. We appreciate the interest in our research shown by the Water Research Institute of the University of Barcelona and by the Department of Algebra, Geometry and Topology of the Complutense University of Madrid.

This research has been supported by the European Research Council (RESCCUE (grant no. 700174)) and the Spanish Ministry of Science, Innovation and Universities (grant no. CGL2017-83866-C3-2-R).

This paper was edited by Alexander Gelfan and reviewed by Hasan Tatli and Serguei G. Dobrovolski.